Among the colony's people suffering under Earth's oppressive rule, a faction plotting to break the status quo through terrorism was born.
Literal
Earth-side [possessive-の] oppression [to-に] suffer colony [possessive-の] people [possessive-の] among [in-で], terrorism [by-による] status-quo-breaking [object-を] scheme force [subject-が] born.
Clearly a sci-fi plot synopsis — Gundam-like setting. 圧政に苦しむ 'suffering under tyranny' is a relative clause modifying コロニーの人々. 現状打破 is a four-kanji compound: 現状 'status quo' + 打破 'breaking through.' 目論む 'to scheme, to plot' is slightly literary. The sentence-final 誕生 without する creates a dramatic, headline-like fragment.